From Feynman-Kac Formulae to Numerical Stochastic Homogenization in Electrical Impedance Tomography

TL;DR

This paper develops a mathematical framework using Feynman-Kac formulae and stochastic homogenization to improve numerical methods for electrical impedance tomography, especially with complex conductivities.

Contribution

It introduces a rigorous derivation of Feynman-Kac formulae for anisotropic conductivities and establishes a foundation for scalable stochastic numerical homogenization schemes.

Findings

Derived Feynman-Kac formulae for anisotropic conductivities

Proved stochastic homogenization for boundary value problems in EIT

Provided convergence estimates for stochastic processes

Abstract

In this paper, we use the theory of symmetric Dirichlet forms to derive Feynman-Kac formulae for the forward problem of electrical impedance tomography with possibly anisotropic, merely measurable conductivities corresponding to different electrode models on bounded Lipschitz domains. Subsequently, we employ these Feynman-Kac formulae to rigorously justify stochastic homogenization in the case of a stochastic boundary value problem arising from an inverse anomaly detection problem. Motivated by this theoretical result, we prove an estimate for the speed of convergence of the projected mean-square displacement of the underlying process which may serve as the theoretical foundation for the development of new scalable stochastic numerical homogenization schemes.